Problem: The product of two consecutive integers is 380. What is the sum of the two integers?
Solution: If two integers are both greater than $\sqrt{380}$, then their product is greater than 380. Similarly, if two integers are both less than $\sqrt{380}$, then their product is less than 380. Therefore, the two consecutive integers that multiply to give 380 are the integers that $\sqrt{380}$ is between. Since \begin{align*} \sqrt{361}<&\sqrt{380}<\sqrt{400} \implies\\ 19<&\sqrt{380}<20, \end{align*} the integers are 19 and 20. The sum of 19 and 20 is $\boxed{39}$.